### Computing Indexes and Periods of All Boolean Matrices Up to Dimension n=8

keywords: Boolean matrix, enumeration, directed graphs

A set of n times n Boolean matrices along with the Boolean matrix multiplication operation form a semigroup. For each matrix A it is possible to find index mu and period łambda, such that A mu = A mu + łambda, and mu, łambda are the smallest positive integers with this property. We are concerned with a question: How many n times n Boolean matrices have the given index, and period? A new algorithm is presented that was used to compute index and period statistics of all square Boolean matrices up to n=8. Computed statistics are presented in the appendix of the paper.

reference: Vol. 31, 2012, No. 6, pp. 1329–1344